10  t-test

• Load packages:

library("readxl")
library("tidyverse")

data_vowels <- read.csv("Vowels_Apache.csv", sep = "\t")

10.1 The \(t\)-test

Since the \(\chi^2\) measure exclusively works with categorical variables, a separate test statistic is required if one of them is a continuous variable. The \(t\) statistic is often used for research questions involving differences between sample means. The way \(t\) is calculated depends on the sources of \(X\) and \(Y\): Are they from the same sample or from two (in-)dependent ones?

First, we consider two independent samples from a population:

• Sample \(X\) with the observations \(x_1, x_2, ..., {x_n}_1\), sample size \(n_1\), sample mean \(\bar{x}\) and sample variance \(s^2_x\).

• Sample \(Y\) with the observations \(y_1, y_2, ..., {y_n}_2\), sample size \(n_2\), sample mean \(\bar{y}\) and sample variance \(s^2_y\).

Definition of the \(t\)-test

The \(t\)-statistic after Welch is given by:

\[ t(x, y) = \frac{|\bar{x} - \bar{y}|}{\sqrt{\frac{s^2_x}{n_1} + \frac{s^2_y}{n_2}}} \]

• If there is more than one observation for a given subject (e.g, before and after an experiment), the samples are called dependent or paired. The paired \(t\)-test assumes two continuous variables \(X\) and \(Y\).

• In the paired test, the variable \(d\) denotes the difference between them, i.e., \(x - y\). The corresponding test statistic is obtained via

\[ t(x, y) = t(d) = \frac{\bar{d}}{s_d} \sqrt{n}. \]

Note the difference \(\bar{d} = \frac{1}{n}\sum_{i=1}^n{d_i}\) and the variance

\[ s^2_d = \frac{\sum_{i=1}^n({d_i} - \bar{d})^2}{n-1}. \]

Traditionally, the \(t\)-test is based on the assumptions of …

1. Normality and
2. Variance homogeneity (i.e., equal sample variances). Note that this does not apply to the \(t\)-test after Welch, which can handle unequal variances.

Implementation in R: Manual vs. automatic

By hand:

# Subset the data by sex
data_m <- data_vowels[data_vowels$SEX == "M", ]
data_f <- data_vowels[data_vowels$SEX == "F", ]

# Compute sample means
mean_m <- mean(data_m$HZ_F1)
mean_f <- mean(data_f$HZ_F1)

# Compute sample variances
var_m <- var(data_m$HZ_F1)
var_f <- var(data_f$HZ_F1)

# Determine sample sizes
n_m <- length(data_m$HZ_F1)
n_f <- length(data_f$HZ_F1)

# Compute t-statistic
t_statistic <- abs(mean_m - mean_f) / sqrt((var_m / n_m) + (var_f / n_f))

# Compute degrees of freedom (simple version)
df <- n_m + n_f - 2

# Find the p-value using the cumulative distribution function (CDF) of the t-distribution
p_value <- 2 * pt(-t_statistic, df)

Or, more concisely:

t.test(data_vowels$HZ_F1 ~ data_vowels$SEX, paired = FALSE) # there is a significant difference!

    Welch Two Sample t-test

data:  data_vowels$HZ_F1 by data_vowels$SEX
t = 2.4416, df = 112.19, p-value = 0.01619
alternative hypothesis: true difference in means between group F and group M is not equal to 0
95 percent confidence interval:
  8.403651 80.758016
sample estimates:
mean in group F mean in group M 
       528.8548        484.2740 

Important

If at least one assumption of the \(t\)-test has been violated, it is important to use a non-parametric test such as the Wilcoxon-Mann-Whitney (WMW) U-Test instead. In essence, this test compares the probabilities of encountering a value \(x\) from sample \(X\) that is greater than a value \(y\) from sample \(Y\). For details, see ?wilcox.test().

10.2 Workflow in R

10.2.1 Define hypotheses

• \(H_0:\) mean F1 frequency of men \(=\) mean F1 frequency of women.

• \(H_1:\) mean F1 frequency of men \(\ne\) mean F1 frequency of women.

10.2.2 Descriptive overview

We select the variables of interest and proceed calculate the mean F1 frequencies for each level of SEX, requiring a grouped data frame.

# Filter data so as to show only those observations that are relevant

data_vowels %>% 
  # Filter columns
  select(HZ_F1, SEX) %>%
    # Define grouping variable
    group_by(SEX) %>% 
      # Compute mean and standard deviation for each sex
      summarise(mean = mean(HZ_F1),
                sd = sd(HZ_F1)) -> data_vowels_stats

knitr::kable(data_vowels_stats)

SEX	mean	sd
F	528.8548	110.80099
M	484.2740	87.90112

# Plot distribution

## Plot means

data_vowels_stats %>% 
  ggplot(aes(x = SEX, y = mean)) +
    geom_col() +
    geom_errorbar(aes(x = SEX,
                    ymin = mean-sd,
                    ymax = mean+sd), width = .2) +
    theme_classic()

## Plot quartiles
data_vowels %>% 
  ggplot(aes(x = SEX, y = HZ_F1)) +
    geom_boxplot() +
    theme_classic()

10.2.3 Check \(t\)-test assumptions

# Normality

shapiro.test(data_vowels$HZ_F1) # H0: data points follow the normal distribution

    Shapiro-Wilk normality test

data:  data_vowels$HZ_F1
W = 0.98996, p-value = 0.5311

# Check histogram

ggplot(data_vowels, aes(x = HZ_F1)) +
  geom_histogram(bins = 30) +
  theme_classic()

# Variance homogeneity

var.test(data_vowels$HZ_F1 ~ data_vowels$SEX) # H0: variances are not too different from each other

    F test to compare two variances

data:  data_vowels$HZ_F1 by data_vowels$SEX
F = 1.5889, num df = 59, denom df = 59, p-value = 0.07789
alternative hypothesis: true ratio of variances is not equal to 1
95 percent confidence interval:
 0.949093 2.660040
sample estimates:
ratio of variances 
          1.588907 

10.2.4 Running the test

# t-test for two independent samples 

t.test(data_vowels$HZ_F1 ~ data_vowels$SEX, paired = FALSE) # there is a significant difference!

    Welch Two Sample t-test

data:  data_vowels$HZ_F1 by data_vowels$SEX
t = 2.4416, df = 112.19, p-value = 0.01619
alternative hypothesis: true difference in means between group F and group M is not equal to 0
95 percent confidence interval:
  8.403651 80.758016
sample estimates:
mean in group F mean in group M 
       528.8548        484.2740 

10.2.5 Optional: Effect size

Cohen’s d is a possible effect size measure for continuous data and is obtained by dividing the difference of both sample means by the pooled standard deviation:

\[\frac{\bar{x} - \bar{y}}{\sqrt{\frac{{(n_1 - 1)s_x^2 + (n_2 - 1)s_y^2}}{{n_1 + n_2 - 2}}}}.\]

library("effsize")

# By hand:
## Compute pooled standard deviation sp
sp <- sqrt(((n_m - 1) * var_m + (n_f - 1) * var_f) / (n_m + n_f - 2))

## Compute Cohen's d
d <- abs(mean_m - mean_f) / sp

# Automatically:
cohen.d(data_vowels$HZ_F1, data_vowels$SEX) # see also ?cohen.d for more details

Cohen's d

d estimate: 0.4457697 (small)
95 percent confidence interval:
     lower      upper 
0.07976048 0.81177897 

10.2.6 Reporting the results

According to a two-sample \(t\)-test, there is a significant difference between the mean F1 frequencies of male and female speakers of Apache (\(t = 2.44\), \(df = 112.19\), \(p < 0.05\)). Therefore, \(H_0\) will be rejected.